Random polytopes in a convex polytope, independence of shape, and concentration of vertices

Write ~.-"d for the set of all convex bodies (convex compact sets with nonempty interior) in ~d. Define o@g~l d as the set of those K E 5 b "~d with vol K = 1. Fix K E .~g-i d and choose points X l , . . . , x~ E K randomly, independently, and according to the uniform distribution on K. Then K,~ = c o n v ( x l , . . . , xn} is a random polytope in K . Write E(K, n) for the expectation of the random variable v o l ( K \ K n ) . E(K, n) shows how well K,~ approximates K in volume on the average. Groemer [Grl] proved that, among all convex bodies K E o@g~l d, the ellipsoids are approximated worst, i.e.

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